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Stability by Fourier series method It is known that most of the elementary functions in mathematics can be written as Fourier series interms of sine and cosine functions, however, the computations involved to find the stability criteria become easier if the complex exponential form is used, i.e., the terms are replaced with the equivalent and l is the length of the domain in the space direction. To avoid any confusion between the complex numbers i and the subscript 'i' until now used to denote node numbers, in this chapter a notation , where p represents the node number in space direction and q represents the node number in time direction, have been used. In the new notation, a discrete function is denoted as ................. (6.4.26) where and h is the step length in x - direction, that is , Dx. [Note that n is the index used to represent the summation in the Fourier series but not the index used to denote the node in time direction, as alerady been stated above q will be used for this purpose in the present section.] Denote the error at the pivotal points along t = 0 between x = 0 and Nh by E(ph)=E_p, for p = 0, 1, 2 . . . .N ................. (6.4.27) (Note that the boundary nodes are also included in (6.4.27) since fourier series method can't incorporate the boundary conditions.) then ................. (6.4.28) System (6.4.28) has (N+1) equations which are sufficient to solve the (N+1) unknowns A₁, A₂, . . . A_n. Since the finite difference equations are always linear, it is sufficient to consider a single term because these solutions are additive. Now to find the propagation of the initial error as t increases, consider the error at any node (p, q) in the form ................. (6.4.29) In (6.4.29) the Fourier summation index n has been dropped because it is considered only one term instead of all n+1 terms. Assume ................. (6.4.30) where k is the step length in time direction that is, is ingeneral a complex constant. Then the error does not increase with q if ................. (6.4.31) Note:This method is applicable to only initial value problems and linear difference equations with constant coefficients. The condition is necessary and sufficient for two time level schemes but is not always sufficient for three or more level finite difference schemes though it is always necessary. Example 6.4.3 Find the condition for stability of the fully implicit finite difference scheme (6.4.6) to one-dimensional diffusion equation scheme (6.4.1). In the new notation is p = 0, 1, 2 . . . N & q = 0, 1, 2 . . . since error equation also satisfies the same finite difference equation we have From (6.4.28) we get since is always positive, the denominator is always greater than one and for all r. Therefore the fully implicit scheme to one dimensional diffusion equation is unconditionally stable.

Stability by Fourier series method

It is known that most of the elementary functions in mathematics can be written as Fourier series interms of sine and cosine functions, however, the computations involved to find the stability criteria become easier if the complex exponential form is used, i.e., the terms are replaced with the equivalent and l is the length of the domain in the space direction. To avoid any confusion between the complex numbers i and the subscript 'i' until now used to denote node numbers, in this chapter a notation , where p represents the node number in space direction and q represents the node number in time direction, have been used. In the new notation, a discrete function is denoted as

................. (6.4.26)

where and h is the step length in x - direction, that is , Dx.

[Note that n is the index used to represent the summation in the Fourier series but not the index used to denote the node in time direction, as alerady been stated above q will be used for this purpose in the present section.]

Denote the error at the pivotal points along t = 0 between x = 0 and Nh by

E(ph)=E_p, for p = 0, 1, 2 . . . .N

................. (6.4.27)

(Note that the boundary nodes are also included in (6.4.27) since fourier series method can't incorporate the boundary conditions.) then

................. (6.4.28)

System (6.4.28) has (N+1) equations which are sufficient to solve the (N+1) unknowns A₁, A₂, . . . A_n. Since the finite difference equations are always linear, it is sufficient to consider a single term because these solutions are additive.

Now to find the propagation of the initial error as t increases, consider the error at any node (p, q) in the form

................. (6.4.29)

In (6.4.29) the Fourier summation index n has been dropped because it is considered only one term instead of all n+1 terms.

Assume

................. (6.4.30)

where k is the step length in time direction that is, is ingeneral a complex constant.

Then the error does not increase with q if

................. (6.4.31)

Note:This method is applicable to only initial value problems and linear difference equations with constant coefficients. The condition is necessary and sufficient for two time level schemes but is not always sufficient for three or more level finite difference schemes though it is always necessary.

Example 6.4.3

Find the condition for stability of the fully implicit finite difference scheme (6.4.6) to one-dimensional diffusion equation scheme (6.4.1).

In the new notation is

p = 0, 1, 2 . . . N & q = 0, 1, 2 . . .

since error equation also satisfies the same finite difference equation we have

From (6.4.28) we get

since is always positive, the denominator is always greater than one and for all r. Therefore the fully implicit scheme to one dimensional diffusion equation is unconditionally stable.